Novel ANN method for solving ordinary and fractional Black-Scholes equation

TL;DR

This paper introduces a novel two-layered ANN approach combined with discretization and optimization techniques to efficiently solve both fractional and ordinary Black-Scholes PDEs, demonstrating improved accuracy and convergence.

Contribution

It presents a new hybrid method integrating discretization, domain mapping, and Adam optimization within an ANN framework for solving fractional and ordinary Black-Scholes equations.

Findings

High accuracy in solving Black-Scholes models

Faster convergence compared to traditional methods

Effective handling of infinite domain issues

Abstract

The main aim of this study is to introduce a 2-layered Artificial Neural Network (ANN) for solving the Black-Scholes partial differential equation (PDE) of either fractional or ordinary orders. Firstly, a discretization method is employed to change the model into a sequence of Ordinary Differential Equations (ODE). Then each of these ODEs is solved with the aid of an ANN. Adam optimization is employed as the learning paradigm since it can add the foreknowledge of slowing down the process of optimization when getting close to the actual optimum solution. The model also takes advantage of fine tuning for speeding up the process and domain mapping to confront infinite domain issue. Finally, the accuracy, speed, and convergence of the method for solving several types of Black-Scholes model are reported.